Clifford theorem
A theorem establishing an inequality between the degree and the dimension of a special divisor on an algebraic curve. It was proved by W. Clifford.
Let be a smooth projective curve over an algebraically closed field, and let
be a divisor on
. Let
be the degree and
the dimension of
. A positive divisor
is called special if
, where
is the canonical divisor on
. Clifford's theorem states:
for any special divisor
, with equality if
or
or if
is a hyper-elliptic curve and
is a multiple of the unique special divisor of degree 2 on
. An equivalent statement of Clifford's theorem is:
, where
is the linear system of
. It follows from Clifford's theorem that the above inequality holds for any divisor
on
for which
, where
is the genus of
.
References
[1] | R.J. Walker, "Algebraic curves" , Springer (1978) |
[2] | N.G. Chebotarev, "The theory of algebraic functions" , Moscow-Leningrad (1948) (In Russian) |
[3] | I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) |
[4] | R. Hartshorne, "Algebraic geometry" , Springer (1977) |
Comments
References
[a1] | P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) |
[a2] | E. Arbarello, M. Cornalba, P.A. Griffiths, J.E. Harris, "Geometry of algebraic curves" , 1 , Springer (1985) |
Clifford theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Clifford_theorem&oldid=23784